function [J, grad] = lrCostFunction(theta, X, y, lambda)
%LRCOSTFUNCTION Compute cost and gradient for logistic regression with 
%regularization
%   J = LRCOSTFUNCTION(theta, X, y, lambda) computes the cost of using
%   theta as the parameter for regularized logistic regression and the
%   gradient of the cost w.r.t. to the parameters. 

% Initialize some useful values
m = length(y); % number of training examples

% You need to return the following variables correctly 
J = 0;
grad = zeros(size(theta));

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost of a particular choice of theta.
%               You should set J to the cost.
%               Compute the partial derivatives and set grad to the partial
%               derivatives of the cost w.r.t. each parameter in theta
%
% Hint: The computation of the cost function and gradients can be
%       efficiently vectorized. For example, consider the computation
%
%           sigmoid(X * theta)
%
%       Each row of the resulting matrix will contain the value of the
%       prediction for that example. You can make use of this to vectorize
%       the cost function and gradient computations. 
%
% Hint: When computing the gradient of the regularized cost function, 
%       there're many possible vectorized solutions, but one solution
%       looks like:
%           grad = (unregularized gradient for logistic regression)
%           temp = theta; 
%           temp(1) = 0;   % because we don't add anything for j = 0  
%           grad = grad + YOUR_CODE_HERE (using the temp variable)
%

% 代价函数向量公式：J = 1 / m * (-y' * log(h) - (1 - y)' * log(1-h)), h = sigmoid(X * theta)

J = 1 / m * (-y' * log(sigmoid(X * theta)) - (1 - y)' * log(1 - sigmoid(X * theta)));

% 梯度向量公式：grad = 1 / m * X' * (h - y), h = sigmoid(X * theta)

grad = 1 / m * X' * (sigmoid(X * theta) - y);

% 代价函数正则化公式(n > 1)：J2 = J + lambda / (2 * m) * sum(theta(n) .^ 2)
J = J + lambda / (2 * m) * (sum(theta(2:end) .^ 2));

% 梯度正则化公式(n > 1)：grad2 = grad + lambda / m * theta(n)
theta_temp = theta;
theta_temp(1) = 0;
grad = grad + lambda / m * theta_temp;

% =============================================================

grad = grad(:);

end
